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For Review OnlyGenetic analysis of growth curve of Afshari lambs by
Legendre Polynomials-based Random Regression Models
Journal: Songklanakarin Journal of Science and Technology
Manuscript ID SJST-2019-0216.R1
Manuscript Type: Original Article
Date Submitted by the Author: 04-Aug-2019
Complete List of Authors: Ghafouri-Kesbi, FarhadEskandarinasab, Moradpasha; University of Zanjan
Keyword: sheep, body weight, heritability, genetic correlation
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Genetic analysis of growth curve of Afshari lambs by Legendre
Polynomials-based Random Regression Models
Farhad Ghafouri-Kesbi1, Moradpasha Eskandarinasab2
1Department of Animal Science, Faculty of Agriculture, Bu-Ali Sina University, Hamedan,
Iran.
2Department of Animal Sciences, Faculty of Agriculture, University of Zanjan, Zanjan, Iran
Corresponding Author: Email address: [email protected], [email protected]
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Genetic analysis of growth curve of Afshari lambs by Legendre
Polynomials-based Random Regression Models
ABSTRACT
The aim of this study was to apply Simple Repeatability Model (SRM) and Random
Regression Models (RRMs) to describe growth curve of Afshari lambs. Results revealed the
inadequacy of SRM to model variation in growth curve of Afshari lambs. A RRM with orders
3, 2, 3, and 2 for direct genetic, direct permanent environment, maternal genetic and maternal
permanent environmental effects was selected as the parsimonious model. Direct heritability
(h2) and maternal permanent environmental effect (c2) were maximum at 98 days of age, and
decreased with age until the end of growth trajectory. Direct permanent environmental effect
(p2) and maternal heritability (m2) were minimum at 98 days of age, but increased thereafter
to a peak at 525 days of age. In conclusion, results revealed substantial genetic potential for
selection responses in early growth of Afshari lambs and that this genetic potential can be
exploited by breeders to improve growth performance of Afshari lambs.
Keywords: sheep, body weight, heritability, genetic correlation
1. Introduction
Body weight is one of the most important economic traits in sheep breeding throughout the
world. Especially in countries where the sale price is based on weight, live weight has a direct
effect on the profitability of the production system. Body weight can be measured at different
points of growth trajectory. Collecting body weights at different ages makes it a typical
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example of so-called longitudinal data (Meyer & Hill, 1997). Analysis of such repeated
records require efficient statistical techniques. Different approaches and models applied to
longitudinal data are reviewed extensively by Lindsey (1993). Among them, Simple
Repeatability Model (SRM), Multi Trait Model (MTM) and Random Regression Model have
been used to genetic analysis of repeated records. A common result which comes from these
papers is the superiority of RRM (see for example Meyer, 2004; Oh, See, Long & Galvin,
2006). Due to this superiority, over the last decade, RR model has been applied for analysis
of repeated records from animal breeding schemes, such as test day milk yield (Schaeffer &
Dekkers, 1994), growth (Rafat et al., 2011), feed intake (Schenkel, Devitt, Wilton, Miller, &
Jamrozik, 2002), egg number (Wolc et al., 2013), fat and mussel depth (Fischer, Van der
Werf, Banks, Ball, & Gilmour, 2006) and total sperm production (Oh et al., 2006).
The Afshari sheep is one of the heaviest breeds of sheep in Iran and is widely distributed
in the Zanjan province. Their population is about 1 million head and mainly farmed for meat
production. This breed is known with appropriate growth characteristics which make them
to be an appropriate breed for selection programs aimed at increasing the efficiency of meat
production (Eskandarinasab, Ghafouri-Kesbi, & Abbasi, 2010). In spite of reports indicating
increase in the genetic evaluation of growth by applying RRM (Meyer, 2004) little efforts
have been made to analysis growth curve of sheep by RRM (Lewis & Brotherstone, 2002;
Fischer, Van der Werf, Banks, & Ball, 2004; Ghafouri-Kesbi, Eskandarinasab & Shahir,
2008). In the current study, therefore, weight records of Afshari sheep from 98 to 525 days
of age were analyzed using SRM and RRM to estimate genetic and non-genetic components
of body weight in this breed.
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2.Materials and Methods
2.1.Data
Body weight records and pedigree information on Afshari lambs were obtained from a
Afshari sheep flock at the department of Animal Science of the Zanjan University, Iran.
Data recorded between 1998 and 2005. Each year natural service is started from September
and continued for 51 days. Each group of 10 ewes is allocated to a fertile ram for 2 or 3
days. This mating system allows the identification of sire and dam of each lamb. Lambing
commences in February. At birth, lambs are weighed and identified to their parents. Lambs
are weaned from their mothers at an average age of 120 days. Animals are kept indoors
from November to March and hand-fed according to NRC (1985). Rams are kept in the
flock for maximum three years and ewes are usually culled after 5 lambing.
Data was monitored several times and incorrect records were removed. Meyer (2001)
showed that the order of polynomial fit require increased when birth weight included in the
data. Also, occurrence of “end effect of polynomials” or “Runge’s phenomenon” is highly
expected by inclusion of birth weight (Meyer, 2005). Therefore, according to Fischer et al.
(2004) suggestion, birth weights were removed from the analyses.
2.2.Statistical analysis
To determine significant fixed effects (year of birth, sex of lambs, birth type and age of
dam at lambing), least square analyses using the GLM procedure of SAS (2004) was fitted
on the data. All these effects were found to be significant (p
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2.2.1. Simple Repeatability Model (SRM)
This model is the simplest model proposed for analyzing repeated records. The
assumption of SRM is that measurements at different ages a realization of the same trait. It
assumes that genetic and phenotypic correlations are of the same magnitude and equal to 1.00
(Meyer & Hill, 1997).
2.2.2. Random Regression Model (RRM)
In RRM, an animal’s breeding value is modeled as a function of a covariate which may
be age in studies of growth trajectory (Meyer, 2005). The RR model for repeated body weight
(including both direct and maternal additive genetic and permanent environmental effects)
could be represented as follows (Kirkpatrick et al., 1990):
ijij
k
mmimijm
k
mimijm
k
mimij
k
mmimijm
mmijij tttttFy
CPMA
)()()()()(1111
0000
3
0
where is the record of animal; is the Legendre polynomials of ijythj thi )( ijm t
thm
age; is the standardized age at recording (between -1 to 1); is the fixed part of the model; ijt ijF
are the fixed regression coefficients for modeling the population mean; , , and m im im im
are the random regression coefficients for direct additive genetic, maternal additive im
genetic, direct permanent environmental and maternal permanent environmental effects,
respectively; , , and are the corresponding order of polynomial for each 1Ak 1Mk 1Pk 1Ck
effect and denotes the residual effect. Several RRM analyses considering different orders ij
of fit for the four random effects were carried out to find the most parsimonious model
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describing the data best. To study the importance of maternal effects, these effects were
excluded from the most parsimonious model and change in LogL was monitored. Residual
variance was modeled with two distinct strategies. In the first strategy, the homogeneity
(constancy) of residual variance was assumed from 98 to 525 days of age and in the second
strategy, residual variance was assumed to be heterogeneous with 14 age classes (one month
each).
The WOMBAT program (Meyer, 2007) was used to analysis the data. Models with
different orders of fit were compared using Akaike’s Information Criterion (Akaike, 1974).
Estimates of variance components were used to calculate coefficient of variations as: CV =
(Houle, 1992). 100 × 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒/𝑚𝑒𝑎𝑛
3. Results and Discussion
Figure 1 shows number of records and unadjusted weights by age (day). A total of 85
ages, ranging from 98 to 525 d, was represented. As shown with an almost linear trend, body
weight tended to increase with age from 27 kg at 98 d to 60.8 kg at 525 d. Mean weights and
SD for the whole period was 39.59 kg and 10.49 kg, respectively. Characteristics of data and
pedigree structures are shown in Table 1. Pedigree included 1593 pedigreed animals of which
1401 individual had recorded body weight.
Several analyses with different orders of fit were tried to find a parsimonious model that
described the data adequately (Table 2). A model fitting legendre polynomials to order k = 3
for all four random effects and fourteen measurement residual variance classes with a total
of 38 parameters to be estimated, was the most complex model fitted in the current study.
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The order of fit was not increased beyond 3 as most of animals (34%) had 3 records. The
SRM was among inefficient models (Model 1). Regarding estimates of residual variances for
fourteen growth phases, SRM resulted in significantly higher residual variances which
showed the inadequacy of this model (Table 3). Similar results have been reported by Arango
et al. (2004). Genetic and environmental components of phenotypic variance are frequently
reported to vary over growth trajectory (Fischer et al., 2004; Ghafouri-Kesbi et al., 2008;
Boligon, Mercadante, Lobo, Baldi, & Albuquerque, 2012) which show the erroneous of the
assumption of SRM which emphasizes on constancy of phenotypic variance and its
constituent components over growth trajectory.
According to logL and AIC values, Model 5 (3,2,3,2) which was able to describe the
covariance structure adequately was selected as the parsimonious model. Estimates of
(co)variances and correlations between RR coefficients for Model 5 are presented in Table
4. The first eigenvalue of covariance functions, i.e., the matrix of covariances among RR
coefficient, dominated throughout and accounted for 79, 100, 90 and 100 per cent of the total
variation for additive, maternal genetic, individual permanent environmental and maternal
permanent environmental effects, respectively. In the study by Ghafouri-Kesbi et al. (2008),
the first eigenvalue accounted for more than 90% of the total variation for direct and maternal
covariance functions. Eigenvalues and corresponding eigen-functions of a covariance
function summarise both the variance and the correlation structure (Kirkpatrick, Hill &
Thompson, 1990) and can be used to predict the effect of selection on the shape of growth
curve. Large eigenvalues reflect large genetic variation in growth curve and the opportunity
for changing the shape of growth curve genetically.
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Considering homogeneity of residual variance instead of heterogeneity of residual
variance significantly increased AIC (15752.54 vs. 15356.10). Residual variance results from
environmental effects on phenotype and includes all the unknown effects affecting phenotype
including important non-additive genetic sources. As animals aged, they may experience
different environmental conditions and therefore residual variance might also vary with age
(Meyer, 2001; Huisman, Veerkamp, & Van Arendonk, 2005; Ghafouri-Kesbi et al., 2008).
The results of logL and AIC showed an improvement in the level of fit when maternal
effects included in the model, in comparison to the model in which maternal effects ignored
(Model 6), in agreement with many reports including Albuquerque and Meyer (2001), Lewis
and Brotherstone (2002) and Ghafouri-Kesbi et al. (2008). As a result, in selection programs
aimed at improving growth performance of Afshari sheep, maternal effects need to be
included in the model to prevent bias in prediction of breeding values.
Estimates of variance components for the ages in the data are shown in Figure 2.
Corresponding estimates of genetic parameters and coefficients of variation are given in
Figures 3 and 4, respectively. Direct additive genetic variance was maximum at 98 d, but
decreased to around 150 d of age and then increased to 300 d of age when it started to
decrease until end of growth trajectory. Direct permanent environmental variance and
maternal genetic variance were minimum at 98 d of age and then increased gradually and
reached the highest value at 525 d of age. Maternal permanent environmental variance
showed a decreasing pattern in the whole period in a way that for higher ages it was almost
zero. The observed patterns for genetic parameters were almost similar to corresponding
variance components, though the trends were not as smooth as those of variances. Heritability
was highest at the beginning, subsequently decreased but yet was higher than 0.3 until 365 d
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when it started a sharp decrease afterward. The observed trend for h2 was in agreement with
Samadi, Hatami, Lavaph, Saadi, and Mohamadi (2013), though it contradicted Lewis &
Brotherstone (2002) and Fisher et al. (2004) who reported an increase in heritability of body
weight with age. However, selection in sheep usually done on early growth traits such as
weaning weight which in Afshari sheep is around 98 to 120 days of age with 0.45 to 0.40
heritability. Likewise, with a maximum value at 98 d of age, CVA had a similar general pattern
with h2. It measures additive genetic variability or “capability to change” in body weight at
different ages. CVA can be high in traits with low heritabilities if contribution of the residual
variance to the phenotypic variance is high (Houle, 1992) and vice versa. Both CVA and h2
guarantee maximum response to selection in early growth of Afshari lambs. With some
fluctuations, direct permanent environmental effect (p2) and corresponding CVP increased
with age throughout the period studied in consistent with other reports (Samadi et al., 2013).
Maternal heritability (m2) and maternal additive genetic coefficient of variation (CVM)
increased with age. In most of examined papers, a diminishing trend for m2 after birth or after
weaning has been reported (Fisher et al., 2004; Safaei et al., 2010). Notable m2 beyond
weaning may be due to carry-over effects from weaning weight (Bradford, 1972; Snyman,
Erasmus, van Wyk, & Olovier, 1995). The trends for maternal permanent environmental
effect (c2) and CVC were decreasing throughout the trajectory and were below 0.05 in the
whole period studied. As a result this component of maternal effects can be ignored from the
genetic evaluation procedure.
Different correlations between body weighs at 98, 189, 294, 399, and 525 d of age are
shown in Table 5. All the correlations decreased steadily with increasing lag in age with a
minimum between body weight at 98 and 525 days of age, a result which has been frequently
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reported (Ghafouri-Kesbi et al., 2008; Safaei et al., 2010; Samadi et al., 2013). The genetic
correlation between 98 d body weight and body weights up to 399 d of age are positive. As
a consequence, selection on body weights around weaning age will change body weight in
the period between 98 to 399 d of age in the same direction. Maternal genetic correlations
between 98-day body weight and body weights taken at higher ages are negative which
indicates that the genes of dams which contribute in milk production have some unfavorable
effect on post-weaning body weights.
4. Conclusions
In conclusion, results obtained here showed the presence of notable genetic variation in
growth curve of Afshari lambs up to 365 d of age that can be exploited for improving growth
performance of Afshari lambs. Simple repeatability model didn’t show an acceptable
performance in analyzing repeated body weight records. Accordingly, SRM is not
recommended for analyzing repeated records of livestock. Body weight around weaning
would be appropriate selection criteria as they are measured early in life, show considerable
genetic variation and have positive genetic correlation with other body weights.
Acknowledgements
We thank H. S. Mohammadi, the manager of the Afshari sheep experimental flock, who
provided us the data used in this study.
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Table 1. Summary of pedigree and data structures of the Afshari sheep N
No. of Animals in the pedigree file 1593
No. of Animals with progeny 575
No. of Animals without progeny 1158
No. of Sires with progeny 47
No. of Sires with progeny and record 29
No. of Dams with progeny 478
No. of Dams with progeny and record 304
No. of Grand sire 32
No. of Grand dam 210
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Table 2. Order of fit for direct (KA) and maternal (KM) genetic,
.animal (KP) and maternal permanent (KC) environmental effects
Model KA KP KM KC Npb LogLc AICd
1 1 1 1 1 18 -7847.33 15722.66
2 2 2 2 2 26 -7712.54 15477.08
3 2 2 3 3 32 -7705.31 15474.62
4 2 3 2 3 32 -7668.66 15401.32
5 3 2 3 2 32 -7646.05 15356.10
6 3 2 - - 23 -7693.250 15432.50
7 3 2 3 2 19 -7857.270 15752.54
8 3 3 2 2 32 -7972.628 16009.26
9 3 3 3 3 38 -8022.51 16121.02
Np: Number of parameters, LogL: Log likelihood function, AIC:
Akaike’s information criterion.
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Table 3. Estimates of error variances for 14 growth phases for Simple Repeatability Model (Model 1), the
parsimonious model (Model 5) and the model with assumption of homogeneity of error variance (Model 7)a
Model E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E12 E13 E14
1 9.30 7.42 7.85 1.18 0.50 7.96 9.30 8.29 8.47 19.07 23.08 23.19 55.97 10.39
5 2.18 2.01 4.55 2.28 1.60 5.32 4.48 3.80 2.57 11.73 12.30 15.26 9.99 9.79
7 4.61 4.61 4.61 4.61 4.61 4.61 4.61 4.61 4.61 4.61 4.61 4.61 4.61 4.61
aE1-E14: Estimates for error variances for 14 growth phases
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Table 4. Estimates of variances (diagonal), covariances (below diagonal), and
correlations (above diagonal) between random regression coefficients and
eigenvalues of coefficient matrix for additive genetic (A), maternal genetic (M),
animal permanent environmental (P) and maternal permanent environmental (C)
effects.
0 1 2 Eigenvalue
A
10.515 -0.0510 -0.4201 12.21 (79%)
-2.1765 2.5522 -0.6895 2.62 (17%)
-3.4652 -0.1261 2.4018 0.64 (4%)
M
4.7060 0.9554 0.9997 6.77 (100%)
3.1048 2.0498 0.9477 0.00 (0.00%)
0.2544 0.1693 0.0153 0.00 (0.00%)
P
30.464 0.5877 32.67 (90%)
7.7120 5.6528 3.45 (10%)
C
0.7515 -0.9991 0.96 (100%)
-0.3913 0.2041 0.00 (0.00%)
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Table 5. Correlations between body weights at different agesa
Age1 Age2 ra rp rm rc rph
98 189 0.663 0.926 -0.848 1.000 0.689
98 294 0.232 0.718 -0.843 1.000 0.423
98 399 0.027 0.516 -0.831 0.999 0.246
98 525 -0.053 0.345 -0.818 0.924 0.161
189 294 0.880 0.928 1.000 1.000 0.754
189 399 0.736 0.802 0.999 0.999 0.57
189 525 -0.287 0.674 0.997 0.926 0.347
294 399 0.957 0.967 1.000 1.000 0.744
294 525 -0.271 0.901 0.998 0.931 0.534
399 399 1.000 1.000 1.000 1.000 1.000
399 525 -0.06 0.982 1.000 0.94 0.664
ara, Direct additive genetic correlation; rp, Direct permanent
environmental correlation rm, Maternal additive genetic
correlation, rc Maternal permanent environmental correlation, rph
Phenotypic correlation
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For Review Only98 112126140154168182196210224238252266280294308322336350364378392406420434448483497511525
0
50
100
150
200
250
300
350
400
0
10
20
30
40
50
60
70
80
Age(days)
Num
ber o
f rec
ords
Body
wei
ght(
kg)
Figure 1. Numbers of records (grey bars) and mean weights (black points) for individual ages.
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98 112
126
140
154
168
182
196
210
224
238
252
266
280
294
308
322
336
350
364
378
392
406
420
434
448
483
497
511
525
0
2
4
6
8
10
12
14Direct additive genetic variance (kg2)
Age(days)
98 112
126
140
154
168
182
196
210
224
238
252
266
280
294
308
322
336
350
364
378
392
406
420
434
448
483
497
511
525
02468
101214
Maternal additive genetic variance(kg2)
Age(days)
98 112
126
140
154
168
182
196
210
224
238
252
266
280
294
308
322
336
350
364
378
392
406
420
434
448
483
497
511
525
00.20.40.60.8
11.21.41.6
Maternal permanent environmental variance(kg2)
Age(days)
98 112
126
140
154
168
182
196
210
224
238
252
266
280
294
308
322
336
350
364
378
392
406
420
434
448
483
497
511
525
05
10152025303540
Direct permanent environmental variance(kg2)
Age(days)
Figure 2. Estimates of variance components
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98 112
126
140
154
168
182
196
210
224
238
252
266
280
294
308
322
336
350
364
378
392
406
420
434
448
483
497
511
525
0
0.1
0.2
0.3
0.4
0.5Direct heritability (h2)
Age(days)
98 112
126
140
154
168
182
196
210
224
238
252
266
280
294
308
322
336
350
364
378
392
406
420
434
448
483
497
511
525
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7Direct permanent environmental effect(p2)
Age(days)
98
112
126
140
154
168
182
196
210
224
238
252
266
280
294
308
322
336
350
364
378
392
406
420
434
448
483
497
511
525
0
0.05
0.1
0.15
0.2
0.25
Maternal heritability(m2)
Age(days)
98 112
126
140
154
168
182
196
210
224
238
252
266
280
294
308
322
336
350
364
378
392
406
420
434
448
483
497
511
525
0
0.01
0.02
0.03
0.04
0.05
0.06
Maternal permanent environmental effect(c2)
Age(days)
Figure 3. Estimates of direct and maternal heritability and direct and maternal permanent environmental effects
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98 112
126
140
154
168
182
196
210
224
238
252
266
280
294
308
322
336
350
364
378
392
406
420
434
448
483
497
511
525
0
0.1
0.2
0.3
0.4
0.5
CVA
Age(days)
98 112
126
140
154
168
182
196
210
224
238
252
266
280
294
308
322
336
350
364
378
392
406
420
434
448
483
497
511
525
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7CVP
Age(days)
98 112
126
140
154
168
182
196
210
224
238
252
266
280
294
308
322
336
350
364
378
392
406
420
434
448
483
497
511
525
0
0.05
0.1
0.15
0.2
0.25CVM
Age(days)
98 112
126
140
154
168
182
196
210
224
238
252
266
280
294
308
322
336
350
364
378
392
406
420
434
448
483
497
511
525
0
0.01
0.02
0.03
0.04
0.05
0.06CVC
Age(days)
Figure 4. Estimates of direct genetic (CVA), direct permanent (CVP), maternal genetic (CVM) and maternal permanent environmental coefficient of variation (CVC).
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